Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol68.Issue3.Dec2000:

Int. J. Production Economics 68 (2000) 229}239

Capacity allocation and outsourcing in a process industry
Ton G. de Kok
School of Technology Management, Eindhoven University of Technology, Room F4, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Received 1 April 1998; accepted 1 November 1999

Abstract
In this paper we consider a production facility producing a uniform product that ships a number of di!erent package
sizes to a (number of) stockpoint(s). The packaging capacity owned by the facility is "nite and the actually used capacity
must be reserved sometime before actual use. Furthermore, the packaging capacity must be allocated among the di!erent
package sizes, such that a target "ll rate for each package size is achieved. We propose two di!erent capacity reservation
strategies, both derived from a periodic review order-up-to-policy. One strategy assumes that excess capacity needs
compared with the owned capacity cannot be "lled and are postponed to the future. The other strategy assumes that
excess capacity needs are outsourced. The objective of the paper is to "nd cost-optimal policies within each of the two
classes of policies and then select the best capacity reservation policy. We present some managerial insights into the
impact of various process and cost parameters and the choice of the capacity reservation strategy. ( 2000 Elsevier
Science B.V. All rights reserved.
Keywords: Capacity allocation; Inventory management; Outsourcing

1. Introduction

In this paper we consider a continuous processing plant, the output of which consists of a uniform
product, which is packaged in barrels of di!erent
sizes before shipment to company-owned geographically dispersed stockpoints. Demand for
packaged products is only occurring at these stockpoints. It is assumed that the uniform product can
be stored in su$cient amounts before packaging so
as to guarantee supply of product to be packaged.
Thus the main problem remaining is the planning
of the packaging department. The planning of the
packaging department can be seen as a hierarchical
planning problem consisting of two phases. In
phase 1 the total packaging capacity in working
hours is determined. This decision is taken a "xed
lead time before actual packaging. This lead time is

required to ensure that the planned capacity, i.e.
number of people, is available in the production
period planned for. This phase is called the capacity
reservation phase. In the second phase the available
capacity is allocated to speci"c barrel sizes in order
to satisfy customer demand. This phase is called the

capacity allocation phase.
In this paper we propose a simple hierarchical
planning framework that ensures that prede"ned
customer service levels are met against minimal
cost. In fact we discuss two di!erent strategies with
respect to the capacity reservation in phase 1 combined with a single capacity allocation strategy in
phase 2. The "rst capacity reservation strategy is to
install a "xed capacity C. If the capacity demand
exceeds C then this capacity demand cannot be
"lled and is postponed to future periods. The second capacity reservation strategy also assumes that

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 3 4 - 6

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T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

a "xed capacity C is available, but in this case
additional capacity can be obtained by hiring additional personnel at some labour agency. The "rst

capacity reservation strategy bu!ers against future
uncertainty through inventory bu!ers and "xed
excess capacity. The second capacity reservation
strategy in addition exploits short-term availability
of external capacity.
Clearly a trade-o! must be made in order to
decide which strategy to employ. For a proper
trade-o! between the two strategies we must compare cost-optimal strategies within each class satisfying the service level constraints. So within each
class we have to consider all costs involved and
then determine the optimal capacity reservation
policy parameters. In both cases the policy parameter to be determined is the "xed capacity C.
However, the cost functions to be minimized under
the two strategies di!er, since only the second strategy implies costs for outsourcing. We show that,
if we employ the outsourcing strategy, then the
optimal C satis"es a Newsboy equation. Through
numerical experimentation we provide managerial
insights as to when to employ which strategy.
The literature on stochastic multi-item "nite
capacity planning problems is rather limited.
Glasserman [1] considers systems consisting of

stockpoints in series, where shipments from
a stockpoint to its successor are constrained by
some upper bound. Such serial systems relate to the
production of a single product consisting of a single
subassembly, which in turn consists of another
single subassembly, etc. Such capacitated systems
do not occur in practice. Glasserman [2] studies
a similar problem to ours. The main di!erence lies
in the fact that a "xed amount of capacity is allocated to each package size. In our model capacity is
allocated based on current capacity requirements,
thereby providing more #exibility. Fransoo et al.
[3] present a single machine multi-product system
with set-up times. The focus of the paper is on the
decision hierarchy leading to an aggregate stochastic planning model to determine optimal cycle
times for each product and a run-out time based
operational planning model to schedule the subsequent production runs. Fransoo's paper di!ers
from ours in that he considers a continuous review
system with lost-sales, whereas we consider a peri-

odic review system with backlogging. The literature

on deterministic "nite capacity problems is huge.
For an introduction to this subject we refer to
Graves et al. [4].
The paper is organized as follows. In Section 2
we describe the model under consideration in detail
and we de"ne the capacity reservation strategies. In
Section 3 we discuss the relation of this model with
multi-echelon models. From this discussion we derive a near-optimal allocation policy. In Section 4
we combine the results of Sections 2 and 3 to derive
expressions for the operating characteristics, i.e.
service levels and costs. Procedures for deriving
cost-optimal strategies are given in Section 5. Using
these procedures we are able to provide in Section 6
managerial insights into the question when to use
which policy and how much "xed access capacity
must be installed. Finally Section 7 discusses
further research opportunities.
2. The model
The production facility under consideration
satis"es the demand for N di!erent package sizes

from stockpoints. In turn these stockpoints satisfy
external customer demand for these package sizes.
Without loss of generality we assume that each
stockpoint stores only one package size. The lead
time from the production facility to stockpoint
i equals ¸ , i"1, 2,2, N. The lead time equals the
i
order ful"lment lead time consisting of order processing, production planning, production and distribution. Let D denote the generic random variable
i
for the demand for package size i in an arbitrary
time unit. We assume that the demand for package
size i in subsequent time periods is i.i.d. Demands for
package size i and j in the same period may be
correlated. Stockpoint i has to achieve a "ll rate b , i.e.
i
b "
:
fraction of demand satis"ed directly from
i
stock on hand at stockpoint i, i"1,

2,2, N.
The stockpoints control their stocks according to
periodic review order-up-to-policies.
De"ne
S "
:
i

order-up-to-level of stockpoint i,

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

R"
:
review period in time units,
D (s, t] "
: demand for package size i in time interi
val (s, t], s)t.
At each review moment the orders from the stockpoints are processed as follows. The aggregate capacity requirements are computed and compared
with the reserved capacity. If the aggregate capacity

requirements are less than the capacity reserved,
then all requirements are satis"ed. Otherwise, the
reserved capacity is rationed among the stockpoints.
Let us de"ne the following generic random
variables:
:
Dc "
i

capacity demand from stockpoints i in
an arbitrary time unit, i"1, 2, 2, N.
:
aggregate capacity demand in an arbitDc "
0
rary time unit,
c "
:
time needed for the production of one
i
package size i, i"1, 2,2, N.

: capacity demand from stockpoint i durDc(s, t] "
i
ing time interval (s, t], i"1, 2,2, N.
Then we have the following relations:
Dc"c D , i"1, 2,2, N,
i
i i
N
Dc " + c D ,
0
j j
j/1
Dc(s, t]"c D (s, t], i"1, 2,2, N,
i
i i
N
Dc (s, t]" + c D (s, t].
0
j j
j/1

Capacity is reserved according to periodic review
echelon order-up-to-policy. The echelon capacity is
de"ned as the sum of all capacity reserved plus all
capacity associated with the inventory positions of
the stockpoints. De"ne
:
capacity order-up-to-level.
Sc "
0
The capacity reservation lead time equals ¸ . This
0
lead time is necessary to schedule the workforce
and plan the production to enquire su$cient uniform product to be packaged. We assume that the
production facility owns a "xed packaging capacity
C. Suppose that the order-up-to-policy prescribes
to reserve a capacity Q .
c
Now we distinguish between two capacity reservation strategies:
(i) Fixed capacity reservation strategy: if Q )C,
c

then reserve Q , otherwise reserve C.
c

231

(ii) Capacity reservation strategy with outsourcing: If Q )C, then reserve Q , otherwise rec
c
serve C and outsource Q !C.
c
It is our objective to select the cost-optimal capacity reservation policy within these two classes. To
formulate an expression for the expected total relevant costs per time unit we introduce the following
parameters:
c "
: cost of owned capacity per time unit,
F
c "
: cost of outsourced capacity per time unit,
O
v "
: value of package size i, i"1, 2,2, N,
i
r"
:
interest rate per time unit.
Note that the interest rate may not only include
cost of capital, but may also account for, e.g., obsolesence risks, storage costs. Hence rv equals the
i
holding cost of package size i per time unit. Furthermore, we assume that the value of the package
size is based on the market price, so that we do not
distinguish between the value of products made by
owned and outsourced capacity. We exclude "xed
costs for outsourcing, although it is quite straightforward to include it into the analysis.
The expression for the expected total relevant
cost per time unit TRC (C, Sc , MS NN ) is given by
0
i i/1
TRC(C, Sc , MS NN )
0
i i/1
"c C#c E[(D c!C)`]f
0
F
O
N
(2.1)
# + rv E[X`(C, Sc , MS NN )],
0
i i/1
j
j
j/1
where

G

f"

1 if excess capacity requirements are
outsourced,
0 otherwise,

: net stock at stockpoint j at
X`(C, Sc , MS NN ) "
0
i i/1
j
the end of an arbitrary review period.
In the sequel we will drop the dependence of
X` and TRC on C, Sc , MS NN . Our objective is to
0
i i/1
j
minimize TRC subject to "ll rate constraints.
Hence, we want to solve the following problem:
min

TRC

C, Sc , MS NN
0
i i/1

s.t.

b *bH.
i
i

232

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

Here bH denotes the target "ll rate. Note that
i
b also depends on the choice of the capacity reseri
vation policy, Sc and MS NN . In the next section
0
i i/1
we derive expressions for TRC and b for the two
i
capacity reservation policies under the assumption
of the so-called Balanced Stock rationing policy
introduced by Van der Heijden [5] for the allocation of reserved capacity among the stockpoints.
Based on these expressions we propose procedures
for determination of the cost-optimal policies.

for the waiting times in a D/G/1-queue. In [6]
a simple and e$cient moment-iteration algorithm
is given that computes excellent approximations for
the "rst two moments of the waiting time of an
arbitrary customer. Hence, we can use this algorithm to computer E[>] and E[>2] where
>"lim
> . Hence, we "nd that under the
n?= n
assumption that the system is in its steady state at
time 0, I` "Sc !>. This result will be employed
0
0,0
when we analyse the capacity allocation strategy.
Let us de"ne CRC(C) as

3. Analysis of the model

CRC(C) "
: expected capacity reservation costs per
time unit.

In this section we derive expressions for longterm average costs and customer service levels for
each of the two strategies proposed. First, we analyse the capacity reservation strategies and thereafter we analyse the common capacity allocation
strategy.

Then it readily follows that

3.1. Capacity reservation strategy with xxed capacity
Under the capacity reservation strategy with
"xed capacity C we have to modify the order-upto-strategy with order-up-to-level Sc . Let us de"ne
0
: echelon capacity of overall system immediI~ "
0,n
ately before review moment n,
: echelon capacity of overall system immediI` "
0,n
ately after review moment n.
Taking into account the "nite capacity C we obtain
the following relation between I~ and I` :
0,n
0,n
I` "min(I~ #C, Sc ).
0
0,n
0,n
minus the total capacity
Since I~ equals I`
0,n~1
0,n
demand during ((n!1)R, nR], we obtain
I` "min(I` !Dc ((n!1)R, nR]#C, Sc ).
0
0
0,n~1
0,n
Now let us introduce the random variable > den
"ned by
> "Sc !I` , n*0.
0,n
0
n
Then it is easy to see that
> "max(0, >
#Dc [(n!1)R, nR]!C),
0
n
n~1
n*0.
Following De Kok [6] we observe that the above
equation is identical to Lindley's integral equation

CRC(C)"c C.
F
3.2. Capacity reservation strategy with outsourcing
Under the capacity reservation strategy with
"xed capacity C and in"nite outsourcing capacity
we are always able to raise the echelon capacity to
its order-up-to-level Sc at review moments. Hence
0
I` "Sc .
0
0,0
Yet the choice of C has immediate cost implications. If C increases then the "xed capacity cost
increases, but the outsourcing cost decreases. It is
easy to see that under an order-up-to-policy the
capacity reserved at review moment n equals
the capacity demand during time interval
((n!1)R, nR]. If this capacity demand exceeds
C we outsource the excess demand, otherwise we
need not outsource. Thus we "nd the following
expression for the capacity reservation costs
CRC(C):
CRC(C)"c C#c E[(Dc ((n!1)R, nR]!C)`].
F
O
0
Note that this expression does not depend on n.
Hence, we "nd under the assumption that the system is stationary at time 0,
(3.1)
CRC(C)"c C#c E[(Dc !C)`].
0,R
F
O
Here Dc denotes the aggregate capacity demand
0,R
for package sizes during an arbitrary review period.
But Eq. (3.1) is related to the well-known Newsboy
equation (cf. [7]). This follows from rewriting the

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

above equation as follows:
CRC(C)"c C#c
F
O
"c C#c
F
O

P
P

=
C
=

(y!C) dF c0,R (y)
D
(1!F c0,R (y)) dy.
D

C
The cost-optimal C follows by setting the "rst
derivative of CRC(C) equal to 0 and veri"cation of
a positive second derivative in the optimum found.
It easily follows that
CRC(C)
"c !c (1!F c0,R (C))
F
O
D
dC
and

Since F c0,R is a monotone increasing function, the
D
second derivative is positive for all C and the optimum CH is found by solving for
c !c
F
(3.2)
F c0,R (CH)" O
D
c
O
which is a Newsboy equation. Note that we implicitly used the fact that the choice of C does not
in#uence the holding costs incurred at the stockpoints, nor the customer service levels. This concludes our analyses of the capacity reservation
strategies.
3.3. Capacity allocation strategy
The capacity allocation strategy is based on Van
der Heijden [5], who proposes a so-called Balanced
Stock (BS) rationing rule for the situation where
a stockpoint has to ration available inventory
among its succeeding stockpoints. Here we are confronted with a similar situation. After reserving
capacity at time 0 we must ration available capacity
among di!erent package sizes at time ¸ . To ration
0
the available capacity we aim at achieving the
order-up-to-level S of package size i for each i. If
i
we assume that
N
Sc " + c S ,
0
j j
j/1

then we can be sure that all available capacity is
rationed among the package sizes. This follows
from the fact that at time ¸ the sum of the reserved
0
capacity at time and all echelon capacities of the
individual package sizes equals I !Dc(0, ¸ ]. The
0
0
target sum of all echelon capacities equals
+N c S . Eq. (3.3) together with I )S implies
j/1 j j
0
0
that the total available echelon capacity is less than
or equal to the target echelon capacity. Hence, no
reserved capacity remains unused. The BS
rationing rule is as follows. Let I be de"ned as
i
I "
: inventory position of stockpoint i after
i
rationing at time ¸ , i"1, 2,2, N.
0
Then we have
p
I "S ! i (Dc(0, ¸ ]#>)
0
i
i c
i
with

d2c (C) dF c0,R (C)
#!1 " D
'0.
dC
dC

(3.3)

233

(3.4)

1
p2(Dc )
i,R
# , i"1, 2,2, N.
p"
i 2+N p2(Dc ) 2N
j,R
j/1
The random variable Dc denotes the capacity
i,R
demand for package size i during an arbitrary
review period. Furthermore, we assume that > is
identically equal to 0 for the capacity reservation
strategy with outsourcing. This BS rationing rule
aims at minimizing the probability of imbalance.
Imbalance occurs when the above allocation rule
leads to negative capacity allocated to one or more
package sizes. Van der Heijden et al. [8] show the
robustness of BS rationing for a wide range of
multi-echelon inventory systems under periodic review order-up-to-polices.
From this expression for I we can derive expresi
sions for both costs and service levels for the di!erent package sizes. It is easy to see that the net stock
X at the end of period ¸ #¸ #R equals
i
0
i
X "I !D (¸ , ¸ #¸ #R], i"1, 2,2, N,
i
i
i 0 0
i
and thus we "nd for the average on-hand stock
(3.5)
E[X`]"E[(I !D (¸ , ¸ #¸ #R])`].
i
i
i 0 0
i
Analogously, we "nd for the "ll rate of package size
i,
b "1![E[(D (¸ , ¸ #¸ #R]!I )`]
i
i 0 0
i
i
!E[(D (¸ , ¸ #¸ ]!I )`]]/E[D ]. (3.6)
i 0 0
i
i
i,R

234

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

Now note that
D (¸ , ¸ #¸#R]!I
i 0 0
i
"D (¸ , ¸ #¸ #R]
i 0 0
i
p
# i (Dc (0, ¸ ]#>)!S ,
0
i
c 0
i
D (¸ , ¸ #¸ ]!I
i 0 0
i
i
p
"D (¸ , ¸ #¸ ]# i (Dc (0, ¸ ]#>)!S .
0
i
i 0 0
k
c 0
i
Thus, we need to know the probability distributions of D (¸ , ¸ #¸ ]#(p /c )(Dc (0, ¸ ]#>)
0
i 0 0
i
i i 0
and D (¸ , ¸ #¸ #R]#(p /c )(DcC , ¸]#>).
i 0 0
i
i i
0
We can easily compute the "rst two moments of
each random variable involved. Furthermore, the
random variables added together are independent
of each other. Hence, it is easy to compute the "rst
two moments of the random variables in (3.5) and
(3.6). As in [8] we can "t mixtures of Erlang distributions to the above random variables so that these
expressions can be easily computed.
3.4. Total relevant costs
From the de"nition of TRC and CRC we "nd
N
(3.7)
TRC(C)"CRC(C)# + rv E[X`(C)].
j
j
j/1
Here we emphasized the dependence of TRC and
CRC on the owned capacity C. We want to minimize TRC subject to the service level constraints
b *bH for all j. It follows from the analysis in
i
j
Section 3.1 that the optimal capacity reservation
policy with outsourcing follows from solving for
CH in the Newsboy equation (3.1) and the associated costs follow from the expressions for
CRC(CH) and E[X`(CH)]. Note that in this case
j
E[X`(C)] is independent of C. However, in the case
j
of the "xed capacity reservation policy E[X`(C)]
j
depends on C through the random variable >. It is
easy to "nd for each C the unique policy satisfying
the service level constraints. De"ne
¹RC(C, bH) "
: expected total relevant cost associated with the unique policy that
satis"es the service level constraints
with owned capacity C.

From numerical experiments we found that
TRC(C, bH) is convex in C under the "xed reservation policy. Thus we can compute the optimal
CH by a simple bisection procedure.
In the next section we provide managerial insights into the problem of choosing the optimal
capacity reservation policy. We observe that the
optimal policy depends on the pdf. of D , as well as
i
on the parameters v , c , ¸ , b , j*1, 2,2, N and
i i i i
¸ , c and c .
0 F
O
We normalize the cost-optimization problem as
follows. De"ne
c"
: cost of owned capacity as a fraction of the cost
outsourced capacity,
a"
: cost of capacity per package size i as a fraction
of the value of package size i in case the optimal capacity reservation policy with outsourcing is applied, i"1, 2,2, N.
Clearly,
c
c" F .
c
O
The meaning of a can be explained as follows.
Let us assume that the optimal capacity reservation
policy with outsourcing is used. De"ne
: cost of capacity per package size, i"1, 2,2, N.
vc "
i
The overall capacity cost per time unit equals CRC,
CRC"c CH#c E[(Dc !CH)`].
0,R
F
O
Since E[Dc] equals the capacity requirements per
i
time unit for package size i, it is good practice to
assume that total capacity cost per time unit associated with package
E[Dc ]
i (c CH c E[(Dc !CH)`]).
size i"
0,R
E[Dc ] F ` O
0
By dividing this cost by the expected demand for
package size i per time unit, we "nd
c
vc" i (c CH c E[(Dc !CH)`]).
(3.8)
i E[Dc ] F ` O
0,R
0
Now we use the de"nitions of a and vc to obtain
i
vc"al , i"1, 2,2, N.
i
i

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

Given a and vc we thus compute v through
i
i
vc
v " i , i"1, 2,2, N.
i
a
In the numerical experiments presented in Section
4 we "rst determine CH for the capacity reservation
strategy with outsourcing given the value of c from
Eq. (3.1). Next we compute vc from Eq. (3.8) and
i
v from Eq. (3.9). Finally, we compute the total
i
relevant costs from Eq. (2.1) (or equivalently Eq.
(3.7)) for both capacity reservation strategies and
compute the optimal CH for both to decide which
capacity reservation strategy is optimal.

4. Managerial insights
In this section we obtain managerial insights into
the impact of a number of process characteristics
on the choice of the capacity reservation policy.
Since we deal with a complex problem with a large
number of process characteristics interacting in
some a priori unknown way, we consider only
marginal e!ects of di!erences in one particular process characteristic. In order to make these marginal
e!ects clear we use the following procedure. For
each value of the process characteristic under consideration we determine all combinations of a and
c values that yield identical total relevant costs for
both capacity reservation policies. Knowing this
iso-cost curve in the (a, c)-plane it is easy to determine the optimal capacity reservation policy for
a given combination of a and c. The impact of
changes in the value of the process characteristics is
determined by comparing the associated iso-cost
curves in the (a, c)-plane.
Note that a denotes the capacity cost per product
as a percentage of the total value of the product and
c denotes the "xed capacity cost per time unit as
a percentage of the outsourced capacity cost per
time unit. It is intuitively clear that the question
whether to outsource or not highly depends on the
values of a and c. If a is extremely small than
capacity costs are only a very small portion of the
overall relevant costs. In that case one should focus
on keeping holding costs as low as possible. This
implies that one should choose for outsourcing. If
on the contrary capacity costs are a large part of

235

the total relevant costs, i.e. a is close to 1, then one
should choose the "xed capacity reservation policy.
Similarly, if c is close to 1, then the cost of outsourcing is only marginally higher than the cost of
owned capacity. In that case outsourcing is costbene"cial. If c is close to 0 than the cost of outsourcing is extremely high, so one chooses the "xed
capacity reservation policy at the expense of higher
holding costs.
Although this reasoning is quite intuitive, it does
not help very much for decision-making in practical situations, because typically in practice the
above extreme (a, c) combinations do not exist. As
indicated above the iso-cost curve in the (a, c)-plane
provides us a powerful decision tool. Especially it is
easy to determine the sensitivity of the decision
taken with respect to errors in the estimation of
a and c.
The Pareto base case: In order to determine iso-cost
curves for relevant practical cases we constructed
a situation that obeys Pareto's law, i.e. a small
number of package sizes accounts for the majority
of turnover. The Pareto base case is described in
Table 1.
For each of the package sizes we assume that the
lead time to the local stockpoints equals two time
units, i.e. ¸ "2 for all i. Furthermore, we assume
i
that the required "ll rate b "0.90. The capacity
i
reservation lead time ¸ equals three time units, the
0
review period is one time unit.
4.1. Impact of interest rates
Using the optimization procedures described in
Section 3 we determine for each (a, c)-combination
the optimal policies within each of the two classes
of capacity reservation policies, that yield the same
Table 1
Demand and capacity usage data for the Pareto base case
i

E[D ]
i

p(D )
i

c
i

1
2
3
4
5

50
30
15
3
2

10
10
10
3
4

1
1.25
1.5
2.5
5

236

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

Fig. 1. Iso-cost curves for di!erent values of the interest rate r.

expected overall relevant costs within a tolerance of
0.005. We varied the interest rate as 0.004, 0.008,
0.016 and 0.032. The interest rates 0.004 and 0.008
are comparable to a 20% and 40% interest rate on
a yearly basis, when the time unit equals one week.
The interest rates 0.016 and 0.032 are comparable
to an interest rate of 20% and 40% on a yearly
basis, when the time unit equals one month.
The resulting iso-cost curve is presented
in Fig. 1.
As expected we "nd that as the interest rate
increases, it becomes more favourable to outsource
capacity, since the cost of holding inventory
increases. The capacity outsourcing policy always
yields minimal holding costs. Another typical phenomenon of the iso-cost curves is the steep ascent
beyond some critical value of c. This indicates the
dominant e!ect on total relevant costs of the cost of
outsourcing capacity as compared with the cost of
owned capacity.
4.2. Impact of variability in demand
Another important process characteristic is
demand variability. The measure we use for de-

mand variability is the coe$cient of variation. The
coe$cient of variation (cv) is de"ned as the quotient of the standard deviation and the mean. To
measure the impact of cv on the iso-cost curves we
simpli"ed the Pareto base case with respect to the
demand and capacity usage data. We consider four
products with E[D ]"100, c "1, i"1, 2, 3, 4.
i
i
Lead time and "ll rate parameters remain unchanged. We assumed r equal to 0.004. We varied
cv as 0.5, 1 and 2. In Fig. 2 we show the resulting
iso-cost curves.
We "nd that the cv has a considerable impact on
the selection of the optimal capacity reservation
policies. As cv increases it becomes more favourable to outsource the excess need for capacity. The
intuitive explanation is probably two-fold. First of
all holding costs increase due to higher safety
stocks to maintain the same service level. Secondly,
with increasing cv of demand for di!erent package
sizes, the cv of the demand for capacity also
increases. If one chooses a "xed capacity reservation strategy then increasing variety implies that
more and more periods of low capacity demand
alternate with some periods with excessive capacity
demand. Such a demand pattern implies either a lot

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

237

Fig. 2. Iso-cost curves for di!erent values of cv.

Fig. 3. Iso-cost curves for di!erent capacity usage pro"les.

of idle capacity or lack of capacity. Thus the "xed
capacity reservation strategy becomes more and
more inadequate with increasing cv.
4.3. Impact of capacity usage distribution
Another process characteristic is the capacity
usage per package size. It is not a priori clear
whether di!erences in capacity usage have an
impact on the choice of the optimal capacity reservation policy. To understand the impact of di!erences in capacity usage we consider the situation

described above, i.e. four di!erent package sizes
each with E[D ]"100 and cv "1. We assume
i
i
that the average capacity usage equals 1. We assume that c "1#d(i!3), i"1, 2, 3, 4. We vary
2
i
d as 0, 0.25 and 0.5. Increasing the value of d implies
greater di!erences in capacity usage. With each
value of d a di!erent capacity usage pro"le is associated. Fig. 3 shows the resulting iso-cost curves.
It is clear that the di!erence in capacity usage
pro"le hardly impacts the choice of the capacity
reservation policy. We have no intuitive explanation
for this robustness phenomenon. This insensitivity

238

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

Fig. 4. Iso-cost curves for di!erent numbers of package sizes.

phenomenon shows even stronger when comparing
the three cases with four package sizes with the case
of one package size having the same aggregate
demand and capacity usage characteristics, i.e.
E[D ]"400 and cv "1, and c "1. Again we
1
1
1 2
"nd more or less the same iso-cost curve. Hence,
the selection of the capacity reservation policy depends on the capacity usage pro"le only through its
aggregate average usage.
4.4. Impact of number of package sizes
Another process characteristic considered here is
the number of package sizes N. Again we assume
identical demand and lead time characteristics for
all N package sizes. Although it is clear that the
larger the N, the larger the average capacity usage,
the iso-cost curves in the (a, c)-plane are a means of
comparing situations with di!erent numbers of
package sizes, since the iso-cost curves do not
change if all demands are multiplied by a "xed
constant. We varied N as 4, 8 and 16. In Fig. 4 we
show the iso-cost curves.
As expected we "nd a similar e!ect as observed in
Fig. 2, where we have shown the impact of demand
variability. Due to the increase in N the cv of the
aggregate demand and aggregate capacity usage
decreases. Due to this decrease in cv the outsourc-

ing option becomes less attractive as N increases.
Note again the steep increase in the iso-cost curve
beyond c equal to 80}90%.
This concludes our discussion of the impact of
di!erent process characteristics on the choice of the
capacity reservation policy. Our intention was to
present some generic managerial insights. Of
course, the "nal choice for one capacity reservation
policy or the other depends on the particular
process characteristics in the situation under consideration.

5. Conclusions and further research
In this paper we studied a continuous processing
plant, the output of which consists of a uniform
product, which is shipped in di!erent package sizes
to one or more, possibly geographically dispersed,
stockpoints. We proposed a simple hierarchical
planning framework that ensures that prede"ned
customer service levels are met against minimal
cost. We found optimal policies within two classes
of simple, practically useful, capacity reservation
policies. These capacity reservation policies are
combined with a single capacity allocation policy.
This allocation policy is the so-called linear allocation policy as described in [8,9]. Diks [9] gives

T.G. de Kok / Int. J. Production Economics 68 (2000) 229}239

evidence that such linear allocation policies are
near-optimal. We used the so-called Balanced
Stock rationing rule, because that policy ensures
a minimal probability of imbalance.
The focus of this paper was on the choice
between two types of capacity reservation policies:
a "xed capacity reservation policy and a policy that
outsources excess capacity demand beyond the
available packaging capacity. Through appropriate
normalization through the introduction of c, the
cost of owned capacity as a percentage of the cost of
outsourced capacity, and a, the cost of capacity as
a percentage of the value of a product, it was
possible to show some generic managerial insights.
The most important observation is that there is
some critical c above which it is bene"cial to outsource capacity and below which it is bene"cial not
to outsource capacity. The value of the critical
c depends on the process characteristics. Qualitative insights are obtained into the impact of various
process characteristics on the choice of the capacity
reservation policy. One of the observations is that
this choice depends mainly on the aggregate capacity usage characteristics.
The analysis presented in this paper can be easily
extended to multi-stage systems, where, after the
capacity allocation phase the packaged products
are shipped via a number of intermediate stockpoints to the stockpoints that satisfy customer demand. This analysis combines the results presented
in this paper with the results in [10] for divergent
multi-echelon systems. The key assumption made
in this paper is the availability of su$cient product
to be packaged. Further research is required to "nd

239

practically useful policies for the situation where
this assumption is no longer valid.
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